I'm using the Euler-Maclaurin formula in a research I'm working on. While brilliant, the elementary proof found here does not give me enough detail.

Specifically, I would like to get an integral-residue kind formula for functions which are continuously differentiable only on open intervals. To be precise:

Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N_f := \sum\limits_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits_0^1 f(t)~ dt ~ . $$

If $f$ is also continuously differentiable on $[0,1]$, then the Euler-Maclaurin formula gives a closed-form expression for $R_f^N$ (which depends on an unknown point). If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval - what can be said about the error term?

More generally speaking, if no such result exists, I'm interested in

1. Generalizations to broader function spaces then that of analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Thanks, Amir

I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me enough detail.

Specifically, I would like to get an integral-residue kind of formula for functions which are continuously differentiable only on open intervals. To be precise:

Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N_f := \sum\limits_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits_0^1 f(t)~ dt ~ . $$

If $f$ is also continuously differentiable on $[0,1]$, then the Euler–Maclaurin formula gives a closed-form expression for $R_f^N$ (which depends on an unknown point). If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval — what can be said about the error term?

More generally speaking, if no such result exists, I'm interested in

1. Generalizations to broader function spaces than that of analytic functions.
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely

Specifically, I would like to get an integral-residue kind of formulas for functions which are continuously differentiable only on open intervals. To be precise:

Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N_f := \sum\limits_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits_0^1 f(t)\, dt \, . $$

If $f$ is continuously differentiable on $[0,1]$, then the Euler Maclaurin gives a precise value for $R_f^N$. If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval - what can be said about the error term?

More generally speaking, if no such result exists, I'm interested in

1. Generalizations to broader function spaces then the analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Most of what I've found were either research papers, or citation of a convenient formula for a specific application. I'm looking for good reviews or textbook chapters which focus on the formula itself.

Thanks

  Amir

I'm using the Euler-Maclaurin formula in a research I'm working on. While brilliant, the elementary proof found here does not give me enough detail.

Specifically, I would like to get an integral-residue kind formula for functions which are continuously differentiable only on open intervals. To be precise:

Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N_f := \sum\limits_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits_0^1 f(t)~ dt ~ . $$

If $f$ is also continuously differentiable on $[0,1]$, then the Euler-Maclaurin formula gives a closed-form expression for $R_f^N$ (which depends on an unknown point). If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval - what can be said about the error term?

More generally speaking, if no such result exists, I'm interested in

1. Generalizations to broader function spaces then that of analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Thanks, Amir

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely

1. Generalizations to broader function spaces then the analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Most of what I've found were either research papers, or citation of a convenient formula for a specific application. I'm looking for good reviews or textbook chapters which focus on the formula itself.

Thanks

Amir

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely

Specifically, I would like to get an integral-residue kind of formulas for functions which are continuously differentiable only on open intervals. To be precise:

Consider $f:(0,1) \to \mathbb{R}$ a continously differentiable function, and define $$R^N_f := \sum\limits_{m=1}^{N} f\left( \frac{m}{N} \right) - \int\limits_0^1 f(t)\, dt \, . $$

If $f$ is continuously differentiable on $[0,1]$, then the Euler Maclaurin gives a precise value for $R_f^N$. If the integral on the RHS exists, but the function is not continuously differentiable on the closed interval - what can be said about the error term?

More generally speaking, if no such result exists, I'm interested in

1. Generalizations to broader function spaces then the analytic functions
2. Generalizations to Lebesgue integrals with respect to other measures.
3. Reminder theorems for continuously differentiable functions.

Most of what I've found were either research papers, or citation of a convenient formula for a specific application. I'm looking for good reviews or textbook chapters which focus on the formula itself.

Thanks

Amir